ADMOS 2023

A new Certified Hierarchical and Adaptive RB-ML-ROM Surrogate Model for Parametrized PDEs

  • Haasdonk, Bernard (University of Stuttgart)
  • Kleikamp, Hendrik (University of Münster)
  • Ohlberger, Mario (University of Münster)
  • Schindler, Felix (University of Münster)
  • Wenzel, Tizian (University of Stuttgart)

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In this talk, we present a new surrogate modeling technique for efficient approximation of solutions and output quantities of parametrized partial differential equations. The model is hierarchical as it is built on a full order model (FOM), reduced order model (ROM) and machine-learning (ML) model chain. The model is adaptive in the sense that the ROM and ML model are adapted on-the-fly during a sequence of parametric requests to the model. To allow for a certification of the model hierarchy, as well as to control the adaptation process, we employ rigorous a posteriori error estimates for the ROM and ML models. The model is therefore able to fulfill fixed or adaptively chosen error tolerances for every requested parameter. In particular, we provide an example of an ML-based model that allows for rigorous analytical quality statements. Numerical experiments showcase the efficiency of our approach in different scenarios, for instance a parameter optimization problem and uncertainty quantification. Here, the ROM is instantiated by Reduced Basis Methods and the ML model is given by a neural network or a VKOGA kernel model. However, a wide range of ML algorithms is applicable within the modeling chain while maintaining the certification of the results.